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Abstract

The Drude model for metal is extended to include complex relaxation rates. As a test for what happens to the surface plasmon resonances with such metals, the lifetime is examined for propagating waves across a single planar metal-dielectric interface. By analytically solving the dispersion relation being fourth-order in the complex frequency, group-velocity dispersion and quality factors are explicitly found. Due to the symmetry breaking between the forward and backward waves, standing waves are not allowed in general.

Figures (5)

(a) The real and imaginary parts of complex conductance for the Drude model according to Eq. (3). (b) A single planar interface between the upper dielectric and the lower metal-like medium. The in-plane wave number is denoted by k directed in the x-direction. Non-zero γr induces cross-interface energy flows, whereas non-zero γi entails an imbalance between the forward (“F”) and backward (“B”) waves along the interface.

The degree of asymmetry e obtained by varying one parameter while fixing the remaining two: e(k) based on the data of Fig. 2(a) with γr as an abscissa; e(γr) based on the data of Fig. 3(a) with k as an abscissa; and e(γi) vs. γi.

(a) Migrations of four roots on the complex (ωr,ωi)-plane. (b) The quality factor plotted against ωr. The curves are generated by varying kover the range 0≤k≤2, where the arrows mean the direction of increasing k. The filled circles in pink color indicate the starting states at k=0, whereas the diamonds indicate the states on the half-way at k=1. In the inset of panel (a), we show the symmetric trajectories for the specified data γr=0.2 and γi=0 (namely, a real-valued γ).

(a) Migrations of four roots on the complex (ωr,ωi)-plane. The arrows imply the direction of increasing γr. The filled circles in pink color indicate the starting states at γr=0 (where ωi=0, thus being neutrally stable), whereas the diamonds indicate the states on the half-way at γr=1.5. In the inset of panel (a), we show the symmetric trajectories for the prescribed data γi=0 (a real-valued γ) and k=0.5. (b) Migrations of four roots on the complex (vgr,vgi)-plane. The thick horizontal arrow in shaded colors indicates the region of the subluminal group velocity, namely, |vgr|<k=0.5.

The real and imaginary parts of k=ω(1+ε)−1ε according to Eq. (13), where both of (kr,ki) are positive. Three values of the complex relaxation rate are examined: (a) γi=0.2−i0.2, (b) γi=0.03−i0.2, and (c) γi=0.2−i.